Submerged Flow Bridge Scour Under Clear Water Conditions

FOREWORD

This study was conducted in response to State transportation departments’ requests for new design guidance to predict bridge contraction scour when the bridge is partially or fully submerged. The study included experiments at the Turner-Fairbank Highway Research Center (TFHRC) J. Sterling Jones Hydraulics Laboratory and analysis of data from Colorado State University. This report will be of interest to hydraulic engineers and bridge engineers involved in bridge foundation design. It is being distributed as an electronic document through the TFHRC Web site at http://www.fhwa.dot.gov/research/.

Jorge E. Pagán-Ortiz
Director, Office of Infrastructure
Research and Development

Notice

This document is disseminated under the sponsorship of the U.S. Department of Transportation in the interest of information exchange. The U.S. Government assumes no liability for the use of the information contained in this document. This report does not constitute a standard, specification, or regulation.

The U.S. Government does not endorse products or manufacturers. Trademarks or manufacturers’ names appear in this report only because they are considered essential to the objective of the document.

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Prediction
of pressure flow (vertical contraction) scour underneath a partially or fully
submerged bridge superstructure in an extreme flood event is crucial for
bridge safety. An experimentally and numerically calibrated formulation is
developed for the maximum clear water scour depth in non-cohesive bed materials
under different approach flow and superstructure inundation conditions. The theoretical
foundation of the scour model is the conservation of mass for water combined
with the quantification of the flow separation zone under the bridge deck
superstructure. In addition to physical experimental data, particle image velocimetry
measurements and computational fluid dynamics simulations are used to
validate assumptions used in the derivation of the scour model and to calibrate
parameters describing the separation zone thickness. With the calibrated model
for the separation zone thickness, the effective flow depth (contracted flow
depth) in the bridge opening can be obtained. The maximum scour depth is
calculated by identifying the total bridge opening that creates conditions
such that the average velocity in the opening, including the scour depth, is equal
to the critical velocity of the bed material. Data from previous studies by Arneson
and Abt and Umbrell et al. are combined with new data collected as part of
this study to develop and test the proposed formulation.

Pressure flow (also known as vertical contraction) scour
occurs when a bridge deck is insufficiently high such that the bridge
superstructure becomes a barrier to the flow, causing the flow to vertically contract
as it passes under the deck. A bridge deck is considered partially submerged
when the lowest structural element of the bridge is in contact with the flowing
water but the water is not sufficiently high to overtop the bridge deck. It is
considered fully submerged when a portion of the flow overtops the bridge deck.

Pressure flow generally only occurs in extreme flood events,
but these types of events are relevant for estimation of scour. When flow is
sufficiently high so that it begins to approach the elevation of the bridge
deck, some of the flow may be diverted laterally to the bridge approaches.
Since the bridge approaches are often lower than the bridge deck, this
diversion may reduce the scour potential under the bridge. Designers must
evaluate the effects of scour under the bridge as well as potential damage
caused by flow diversion.

An experimentally and numerically calibrated scour model was developed
in this study
to calculate the maximum clear water scour depth in non-cohesive bed materials under
different deck inundation conditions. The theoretical formulation of the model
is based
on the conservation of mass of the water passing underneath the bridge deck.
Particle image velocimetry (PIV) measurements and computational fluid dynamics
(CFD) simulations were used to validate assumptions used in the derivation and
verify calibration of parameters
included in the scour model. As one of the important parameters in the pressure
flow scour model, the separation zone thickness in the bridge opening was formulated
analytically, calibrated experimentally, and verified by PIV and CFD analyses.
The maximum scour depth was calculated by identifying the total bridge opening
that resulted in the average velocity
in the opening that is equal to the critical velocity of the bed material.

Experimental data were developed for this study at the
Federal Highway Administration's (FHWA) Turner-Fairbank Highway Research Center
(TFHRC). In addition, data from previous studies by Arneson and Abt as well as Umbrell
et al. were retrieved to support the evaluation of the proposed pressure flow
scour model.(1,2)

This report summarizes a literature review on pressure scour
and describes the physical and theoretical foundation for the model formulation.
The newly collected flume data as well as PIV and CFD analyses are summarized
along with the data collected by Arneson and Abt as well as Umbrell et al.(1,2) The model formulation is refined, tested, and compared to other approaches used
to estimate pressure scour. Recommendations for model application are also provided

Investigations on submerged flow bridge scour have been
reported by Arneson and Abt, Umbrell et al., and Lyn.(1-3) Arneson
and Abt conducted a series of flume tests at Colorado State University and
proposed the regression equation in
figure 1.(1)

Vb =
Average velocity of the flow through the bridge opening before scour occurs,
ft/s.

Vc = Critical velocity of the bed material in the bridge
opening, ft/s.

Vc is defined by Arneson and Abt as shown in
figure 2 as follows:(1)

Figure 2. Equation. Critical velocity per Arneson and Abt.(1)

Where:

g = Gravitational
acceleration, ft/s2.

s = Specific
gravity of sediment, dimensionless.

D50 = Median diameter of the bed materials, ft.

Umbrell et al. conducted a series of flume tests in the FHWA
TFHRC J. Sterling Jones Hydraulics Laboratory.(2) Using the mass
conservation law and assuming that the velocity under a bridge at scour
equilibrium is equal to the critical velocity of the upstream flow, they presented
the equation in figure 3.

Vu must
be less than or equal to Vc to insure the clear water scour assumption. When flow does not overtop the bridge
deck (i.e., partially submerged flow), then hw equals zero.

Umbrell et al. modified the previous equation to improve the
fit to their laboratory data, resulting in the equation in
figure
4.
As part of the refinement in this equation, Vc is estimated by the equation in
figure
2,
but the coefficient 1.52 is replaced by 1.58.

Lyn reanalyzed the data collected by Arneson and Abt and
Umbrell et al.(1-3) Lyn identified concerns related to spurious correlation
in the regression for the Arneson and Abt equation and the low quality of
Umbrell et al.'s dataset. He proposed the empirical power law formulation in
figure 5.

Figure 5. Equation. Lyn equation for maximum equilibrium scour.(3)

The equation in
figure 1 is currently recommended for use in the FHWA guidance document for bridge scour
at highway bridges.(4) Concerns raised by Lyn suggest that an improved
model for pressure scour is needed.(3)

Experiments show that scour is generally greatest near the
downstream end of the bridge deck. This observation is commonly attributed to
the vertical contraction (concentration) of flow in the bridge opening. The
flow separates from the leading edge of the bridge deck, creating a flow
separation zone (ineffective flow area) and forcing the flow in the bridge
opening to contract and accelerate. High velocities in the bridge opening
initiate sediment movement and scour.

Pressure flow (vertical contraction) scour may be analyzed by
estimating the effective depth
in the bridge opening for critical velocity to occur at equilibrium scour. The
effective depth (contraction depth) is determined based on the opening that
allows stream flow to pass through at the location of maximum scour.
Figure
6
illustrates pressure scour and defines key parameters including ys and effective depth at the
point of maximum scour, y2.

Figure 6. Illustration. Parameter definitions at
maximum scour.

If hu is
greater than hb, then
there is a vertical contraction of flow that may result in scour.
Figure 6 illustrates a fully submerged bridge deck where flow overtops the bridge. There
is a stagnation streamline that represents the division of the approach flow
between that which passes under the bridge and that which overtops the bridge. The
effective approach flow depth, hue,
represents the portion of the approach flow that is directed under the bridge. For
partially submerged bridge decks, the bridge deck blocks the approach flow, but
there is no weir flow over the deck, and all of the approach flow passes under
the bridge.

At section C in
figure
6,
separation zone thickness, t, is
indicated where there is effectively no flow conveyance. The flow field is
effectively conveyed through the opening represented by the sum of flow
contraction depth at the point of maximum scour, hc, and ys.

Figure 7 illustrates velocity distributions at three section locations. At the approach
section A, the velocity distribution is unaffected by the bridge. For the fully
submerged case shown in the figure, a portion of the approach flow will overtop
the bridge, and a portion will flow beneath the bridge. The flow at the bridge
opening, which is shown in section B, is non-uniform, which indicates a
velocity peak (highest velocity) just below the flow separation zone. The
velocity profile at the point of maximum scour, which is shown in section C, is
also non-uniform, which indicates the possibility of minor backflow in the
separation zone. For analytical purposes, actual velocity distributions are
simplified to average velocities, as shown in
figure 8.

As shown in
figure
7,
shear stress applied by the flowing water, τ o, is
less than the critical shear stress,
τ
c, in the
approach, as is required for clear water approach conditions. However, if there
is sufficient contraction, this relationship will reverse at section B,
initiating scour. At section C, a scour hole will form to increase conveyance
until the applied shear is less than or equal to
τ
c.

Figure 7. Illustration. Velocity distributions.

Figure 8. Illustration. Average velocities.

Conservation of Mass

Contraction of the flow field is created by a partially or fully
submerged bridge superstructure because of the physical blockage of the
superstructure resulting in contraction of the flow. The contraction creates a
separation zone that starts from the lower corner of the leading edge of the
bridge superstructure and increases to a maximum t. Determination of the flow separation thickness is considered
critical to understanding the pressure flow scour mechanism.

Analysis of pressure flow scour focuses on the orifice flow
under the bridge as described in
figure
6.
By the principle of conservation of mass, the discharge rate for the orifice
flow must remain the same for sections A, B, and C. The discharge for unit
width of channel is given by the equation in figure
9.

Figure 9. Equation. Unit discharge.

Where:

q = Unit
discharge, ft2/s.

V = Average
flow velocity, ft/s.

h = Flow depth, ft.

The zone of separation near the lower flange of the girders
is filled with vortices, and the average velocity in this zone is nearly zero. In
figure
6,
section A is located sufficiently upstream so that it is free of the influence
from the bridge superstructure. hue is measured from the stagnation point elevation to the unscoured stream bed. Section
B is located at the leading edge of the bridge superstructure and is considered
to be the beginning of the separation layer. The depth of the flow in section B
is hb. Although some
pressure scour may occur here, it is not sufficiently significant to add to the
conveyance through this section. Maximum scour is located at section C. It is
assumed that the maximum scour occurs under the bridge rather than beyond the
trailing edge of the bridge superstructure. At this section, the separation
zone has increased to a thickness, t,
which effectively reduces the portion of the flow field providing conveyance to hc. Simultaneously, ys effectively increases the
depth of the channel, making the conveyance depth at this section equal to hc plus ys.

Therefore, using the conservation of mass at sections A, B,
and C and recognizing that at equilibrium the velocity at section C is equal to Vc, the equation in
figure 10 is created.

Figure 10. Equation. Conservation of mass.

Where:

Vue = Effective approach velocity directed under the
bridge, ft.

Considering only the continuity between sections A and C and
recognizing that hc = hb - t, the equation in
figure 11 is created.

Figure 11. Equation. Conservation of mass between sections A and C.

When the velocity in the contracted section C is less than
the critical velocity of the bed material, scour will stop (or not begin).
Conversely, when the velocity in the contracted section is greater than
critical velocity, scour will occur, and the depth of scour will increase until
the conveyance increases to the point where the velocity is reduced to the
critical velocity. This occurs at the equilibrium scour depth. Laursen's
relation for critical velocity is shown in figure 12.(5)

Solving this for equilibrium scour results in the equation in figure 14. The first term on the right-hand side of the equation is y2, as shown in figure 6.

Figure 14. Equation. Equilibrium scour.

This relationship reveals that the equilibrium depth of scour
is a function of the unit discharge through the bridge opening, Vuehue, D50, and t. By inspection, scour will only occur when the equation in
figure 15 is satisfied.

Figure 15. Equation. Threshold for scour.

The values of hb and D50 are determined
from the site geometry and sampling of bed
materials, respectively. Determination of the effective approach flow
conditions and t require further consideration.

Effective Flow

The depth of the effective approach flow, hue, requires a determination
of the location of the stagnation streamline. The location of the streamline
will vary, but it is expected to be between the lowest point on the bridge superstructure
and near or slightly above the top of the bridge railing. The location of the
stagnation streamline is attributed to a number of factors, which may include
the shape of the deck, pitch angle of the deck (super elevation), inundation
depth, and weir flow depth. The presence of railing and size of openings in the
railing may also affect the stagnation point.

Consider the case where the approach flow elevation does not
overtop the bridge but any further increase in depth will result in
overtopping. At this level of incipient overtopping, all discharge goes under
the bridge deck, and the stagnation line is effectively at the top of the
bridge superstructure. When the upstream depth exceeds the top of the
superstructure, a small weir flow occurs. With negligible weir flow, it is
reasonable to assume that for this boundary event, the stagnation level is very
close to the top of the bridge superstructure. However, as the approach depth further
increases, it is expected that the stagnation level will gradually move to some
intermediate level on the side of the bridge superstructure as it splits the
flow to weir flow above the bridge and orifice flow below the bridge. In most
cases, the blocked flow directed below the bridge will be larger than the
blocked flow over the bridge, allowing the hypothesis that the stagnation level
will be located in the upper half of the superstructure.

Given the broad range of possible situations, there is no
known theoretical means for determining the general separation streamline
location. This is explored further with the experimental tests, but for the
initial formulation of the scour model, it is assumed that the separation
streamline is located at the top of the bridge superstructure as shown in
figure
6.

With this assumption, the approach flow depth directed under
the bridge is defined by the equation in
figure 16.

Figure 16. Equation. Estimate of effective approach flow depth.

Where:

ht = Flow depth above the bottom of the bridge
superstructure, ft.

For partially submerged flow, ht is less than or equal to the bridge superstructure
thickness, T, and for partially
submerged flow, ht is less
than or equal to T. For fully submerged
flow, hue represents the
portion of the approach flow that will be directed under the bridge deck. This has
been defined as the distance between the stagnation streamline and the
unscoured streambed. As previously stated, the stagnation streamline is conservatively
assumed to be at the top of the superstructure. Therefore, the effective
approach depth for fully inundated flow is estimated by the equation in figure 17.

Figure 17. Equation. Estimate of hue for fully inundated flow.

The effective approach velocity for Vue must also be estimated. For a partially submerged
deck, all flow is directed under the bridge, and Vue is equal to Vu.
For the fully submerged situation, Vue is estimated using the
power law velocity profile as shown in the equation in figure 18.

Figure 18.
Equation. Power law estimate of effective approach velocity.

Where:

n = Exponent used for the power law velocity distribution.

For fully developed turbulent flows, n may be taken as one-seventh. In the field, the assumption of
fully developed turbulent flow is reasonable, but the same is not always true
for laboratory (flume) data. A comparison between a fully developed profile and
a uniform profile is shown in
figure 19.
Because of this difference in velocity profile, the power law exponent should
be taken as zero for experimental data.

The remaining parameter in the equation in
figure 14 is t. Its value is not readily
available from experimental data; therefore, it is necessary to develop an
analytical model that ties t to other
measured parameters. t is potentially
affected by several factors, including the superstructure geometry, vertical
flow contraction, bridge opening size, and approach flow conditions.

The relative importance of these parameters in determining t varies depending on the relationship
between the bridge superstructure and the two flow boundaries-the channel
bottom and the water surface. Therefore, it is useful to consider four
conditions characterized by the relative position of bridge superstructure, the
channel bottom, and the water surface. Assuming a fully submerged bridge
superstructure, the conditions are as follows:

In the first condition, significant overtopping occurs,
and the superstructure is relatively close to the stream bed. Conveyance over
the bridge is large compared to the constriction caused by the superstructure. The
bottom of the superstructure is sufficiently close to the channel bottom, and
the distance between the two may be relevant to t. This condition is referred to as "surface-far-field."

In the second condition, moderate to insignificant
overtopping occurs, and the superstructure is well above the stream bed.
Conveyance under the bridge is large compared to the constriction caused by the
superstructure. However, the proximity of the water surface to the separation
zone is important in t. This condition
is referred to as "bed-far-field."

In the third condition, significant overtopping occurs,
and the superstructure is well above the stream bed. Conveyance both above and
below the bridge is significant compared to the constriction caused by the
superstructure. In this case, neither the water surface nor channel bed
influences t. This condition is
referred to as "far-field."

In the fourth condition, moderate to insignificant
overtopping occurs, and the superstructure is relatively close to the stream
bed. The constriction caused by the superstructure is significant compared to the
conveyance both over and under the bridge. Both the water surface and channel
bed influence t. This condition is
referred to as "shallow water."

Because bridges are generally designed to pass large flood
flows (i.e., the 100-year event) without overtopping, for most cases where
overtopping occurs, it is moderate to insignificant. Therefore, conditions 1
and 4 are of practical concern. Considering the partially submerged
superstructure, conditions 1 and 3 do not exist by definition, leaving only
conditions 2 and 4
for consideration.

For condition 2, t is a function of the characteristics of water, the extent of overtopping, and
the constriction imposed by the superstructure. Conceptually, the equation in
figure 20 describes a function that relates relevant parameters to t.

Figure 20. Equation. Parameters relevant for t.

Where:

μ = Viscosity of water, lb-s/ft2.

ρ = Density of water, slug/ft3.

qB = Unit discharge blocked by the bridge superstructure,
ft2/s.

The unit discharge blocked by the bridge superstructure is
based on Vu and the
physical blockage of the superstructure. For submerged flow, qB is defined in figure 21.
For partially submerged flow, it is defined in figure 22.

For submerged flow, ht is greater than T, and for partially
submerged flow, ht is less
than or equal to T. With all
potential variations of bridge geometry, general geometrical characterization
of the bridge upstream face is a complex task. It is assumed that the tested
bridge shapes are fairly representative for the largest population of typical
bridges.

For submerged flow, the parameters described in figure 20 were configured into dimensionless ratios as shown by the equation in figure 23.

The only difference between the fully and partially submerged conditions is presence of the ratio of T to ht. For the fully submerged case, ht is the sum of T and hw, leading to the relationship shown in figure 27.

Figure 27. Equation. Submerged depth relationship.

Recognizing that in the partially submerged case, hw is zero, the ratio in
figure 27 will always be equal to 1 under such conditions. Therefore, both the fully and
partially submerged cases are represented by
figure 28.

Figure 28. Equation. Unified dimensionless parameter ratios.

Considering the shallow water condition (condition 4),
additional parameters have an effect
on t, including the clearance between
the bottom of the bridge superstructure and hb.
Unlike condition 2, the bridge superstructure is sufficiently close to the
stream bed so that the relative constriction influences t. By observation from experiments and CFD simulations, this proximity
makes the separation zone thinner. Such an effect is more pronounced when the deck
is very close to the bed.

For the shallow water condition, two cases must be satisfied.
First, t should approach zero when
the bottom of the superstructure approaches the water surface elevation. That
is, t should approach zero when ht approaches zero. Second, t should approach zero when the bottom
of the superstructure approaches the stream bed. That is, t should approach zero when hb approaches zero. While the latter limit is informative, such a situation is not
realistic because a bridge would not be constructed so close to the stream bed
and the scour depth would not necessarily approach zero in the limit. Nevertheless,
these two limits can be expressed as dimensionless ratios by dividing by hu and supplementing the
parameters in
figure 28 with these additional terms to develop the proposed equation for t in
figure 29.

The variation of K with
different geometry is beyond the scope of this study. A few typical
deck-and-stringer sections were selected and used for both experimental and CFD
studies.

The two components of the first term of the equation in
figure 29 can be separated as shown in
figure 30.

Figure 30. Equation. Revised equation form for t.

Chapter 4. Experimental Study

New pressure flow scour experiments were conducted at FHWA's TFHRC
J. Sterling Jones Hydraulics Laboratory to further develop and test the
hypotheses proposed in this study. In addition, data from Arneson and Abt as
well as Umbrell et al. were retrieved and used in this study.(1,2) In
addition, a series of CFD simulations were conducted to expand the scope of the
experimental data.

Physical experiments were conducted in a flume under
controlled flow conditions for two uniform bed materials and two bridge deck
models. The facilities, instrumentation, experimental setup, and procedure are described
in this section. The experimental flume was 70 ft long, 6 ft wide, and 1.8 ft
deep with clear sides and a stainless steel bottom with a slope of 0.0007
percent (see
figure 31 ).
A test section that consisted of a narrowed channel that was 10 ft long and
2.07 ft wide and had a 1.3-ft sediment recess was installed in the middle of
the flume. A model bridge was installed in the narrowed section above the
sediment recess. A honeycomb flow straightener and a trumpet-shaped inlet were
carefully designed to smoothly guide the flow into the test channel. Water was
supplied by a circulation system with a sump of 7,400 ft3 and a
pump with capacity of 10.6 ft3/s with the flow depth controlled by a
tailgate. The discharge
was controlled by a LabViewTM program and checked by an
electromagnetic flow meter.

The effect of bridge girder configuration was examined using
a three-girder deck and a six-girder deck as shown in
figure 32.
The bridge model was located at eight different elevations. Both decks had adjustable
rails that could pass overflow on the deck surface, as shown in
figure 33,
permitting the deck to have eight different inundation levels.

Figure 32. Illustration. Bridge deck models (inches).

Figure 33. Illustration. Bridge rail (inches).

A LabViewTM program was used to control an
automated flume carriage that was equipped with a micro-acoustic doppler velocimeter
(MicroADV) for records of velocities and a laser distance sensor for records of
depths of flow and scour. The MicroADV by SonTek® measures three-dimensional
(3D) flow in a cylindrical sampling volume of 0.177 inches in diameter and
0.220 inches in height with a small sampling volume located about 0.2 inches
from the probe.(6) The range
of velocity measurements is from about 0.0033 to 8.2 ft/s. For these
experiments, velocity measurements were taken in a horizontal plane located at
a cross section 0.72 ft upstream of the bridge model. The LabViewTM program was set to read the MicroADV probe and the laser distance sensor for 60
s at a scan rate of 25 Hz. According to Acoustic
Doppler Velocimeter Technical Documentation, Version 4.0, the MicroADV has
an accuracy of ±1 percent of measured velocity, and the laser distance sensor has an accuracy
of ±0.00787 inches.(6)

Two discharges were applied. They were determined by the critical
velocity and the flow cross section in the test channel with a constant flow depth
of 0.82 ft. The critical velocity was preliminarily calculated by Neill's
equation and was adjusted downward by approximately 10 percent for the flow velocity to insure clear water scour.(7) The
upstream velocity for the smaller of the two bed materials was approximately 1.3
ft/s, and the corresponding discharge
was 2.28 ft3/s. With a Reynolds number of 57,000 and Froude number
of 0.17, this approach flow was subcritical turbulent flow. The upstream
velocity for the larger bed material was approximately 1.74 ft/s, and the
corresponding discharge was 2.95 ft3/s. With a Reynolds number of 73,700
and Froude number of 0.22, this approach flow was also subcritical turbulent
flow. The experimental conditions are summarized in
table
3
in the appendix.

The experimental procedure is as follows:

Fill the sediment recess with sand and evenly
distribute the sand on the bottom of the flume until the depth is 2 ft in the
sediment recess and 0.66 ft in the test channel.

Install a bridge deck at a designated elevation and
position it perpendicular to the direction
of flow.

Pump water gradually from the sump to the flume to the
experimental discharge that is checked with the electromagnetic flow meter.

Run each test for 36 to 48 h and monitor scour
processes by grades in a clear side wall.
An equilibrium state is attained when scour changes at a reference point are less
than
0.0394 inches continuously for 3 h.

Gradually empty water from the flume and scan the 3D
scour morphology using the laser distance sensor with a grid size of 1.97 by
1.97 inches.

PIV was employed to assist in the evaluation of t in a separate set of experiments in a
smaller flume. A PIV system generally includes a laser emitter, charge coupled
device camera, and reflective particles that follow the flow. The flow is first
seeded with PIV particles (median diameter of 3.15 mil) upstream of the test
section. A laser beam is spread into a light sheet by a cylindrical lens and
illuminates the particles that travel across the width of the light sheet.

Two images are taken by the camera with a short time delay.(8) The
recorded images are then divided into small interrogation windows. The displacement
of the particles in each interrogation window and subsequently the velocity can
be obtained using cross correlation techniques. The bridge deck used in the PIV
experiments is 0.583 ft wide, 0.909 ft long, and 0.148 ft high. The height of
the girders is 0.066 ft.

CFD simulation was used to extend the laboratory data
collected as part of this study. The CFD models of the flume were calibrated to
the physical model runs, and additional scenarios could be generated. STAR-CCM+
CFD software was installed at the Transportation Research and Analysis
Computing Center at the Argonne National Laboratory and was used to conduct
simulations to estimate the boundary layer thickness and analyze the stagnation
point of deck blockage for partially and fully submerged bridge deck flow
conditions.(9) Flow fields adjacent
to a representative bridge deck for unscoured and scoured bed were simulated. The
six-girder bridge deck used for CFD simulations had the same dimensions as the
model bridge deck used for the flume tests.

Arneson and Abt's experiments were conducted at Colorado
State University in a 300-ft-long
by 9-ft-wide flume with a bed slope adjustable up to 2 percent.(1) Maximum flow circulation capacity was 100 ft3/s. Data were collected
with various flow conditions, four separate bed materials, and six model bridge
deck elevations. Arneson and Abt conducted a total of 72 tests. The
experimental setup, measurements, and data description are detailed in Transportation Research Record1647.(1) A summary of the
test conditions is presented in
table
4
of
the appendix.

For the purposes of this study, some data points collected by
Arneson and Abt were not considered. For the current study, data were required
to meet the following criteria:

Clear water scour occurred when Vu was less than Vc,
as determined by Laursen's critical velocity equation shown in
figure 12 using D50 as the
characteristic sediment size.(5) Test runs with maximum scour less
than 0.394 inches were discarded because the results would not be considered
meaningful by introducing spurious correlations.

The experiments of Umbrell et al. were conducted at TFHRC in
the same flume shown in figure
31
.(2) Data were collected with various flow conditions, three separate bed materials, and six model bridge deck elevations. Umbrell et al. conducted a total of 81
tests. The experimental setup, measurements, and data description are detailed
in "Clear-Water
Contraction Scour Under Bridges in Pressure Flow."(2) A summary of
the test conditions
is presented in
table 5
of the appendix.

As with the Arneson and Abt data, some data points collected
by Umbrell et al. were not considered. Data were required to satisfy the following
criteria:

CFD and PIV analyses were used to assist in the
characterization of the separation zone.
Figure 34 and
figure 35 provide examples of the CFD and PIV simulations, respectively. Scour rate was initially
highest near the bridge opening, but as the scour hole developed, the deepest
scour and largest t moved close to
the downstream end of the bridge opening, as shown in
both figures.

CFD simulations revealed that t was consistent with the model length scale. That is, the ratio of
the first and second exponents in
figure 30, m/n, was 2. Ignoring the slight
geometric difference of deck models used in the experimental program, this was
confirmed in the curve fitting for the values of K and the exponents shown in
figure 36.

It should be noted that within the range of 41 to 68 ·F, the viscosity of
water decreased by approximately one-third. A temperature of 68 ·F was assumed for
the laboratory data used in
this study.

Figure 36.
Equation. Best fit t equation.

Using
figure 36 in conjunction with
figure 14,
the scour depth can be calculated. This computation, using the laboratory data
and normalized by upstream approach depth, is summarized in
figure 37.
Since the exponents and coefficient for estimating t were fit to these data, the fit is relatively good.

Figure 37. Graph. Scour comparison with best fit equation for t.

Because of the complexity of the equation for t, further assessment is desirable.
Given that the ratio of the exponents on the first and second terms in
figure 36 is 2, the effect of ht in
the first term cancels out the effect of that variable in the second term.
Therefore, any length parameter may be substituted without changing the result
of the equation. Substituting hu for ht results in the
equation in
figure 38.
The first term in this equation is the approach Froude number, and the second
term is the approach Reynolds number.

Figure 38. Equation. Best fit t with approach
depth.

The estimation of t in figure 38 can be simplified by multiplying both sides of the equation by ht/hb, yielding
the equation in figure 39.

Figure 39. Equation. Best fit t with
superstructure height.

However, CFD tests revealed behavior of t that was inconsistent with the formulation provided in
figure 39.
First, these tests indicated that t increased
with approach depth for partially submerged conditions but became nearly
constant after the bridge was fully submerged.
In contrast, the last term of the equation suggests that t would continue to increase.
Figure 40 illustrates CFD simulations that show the increase in t with increasing approach depth. In contrast,
figure 41 shows that as the partial submergence approached the top of the superstructure
and became fully submerged, t was nearly
constant. (Note that both figures
show simulations with an approach velocity of 4.9 ft/s.) The cases are for a
fixed bed, and, consequently, the thickest point of the separation zone is
close to the upstream opening of the flooded bridge. The fixed bed imposes some
constriction to the separation zone, making it narrow down and reattach sooner
than when there is a scour hole. It was found that the initial expansion of the
separation zone was not affected by this constriction. Therefore, an exponent
closer to zero for this term is considered for the design equation, changing it
from -0.3 to -0.1.

Figure 40. Illustration. t with depth (partially
submerged).

Figure 41. Illustration. t with depth (mostly
and fully submerged).

The first two terms of the equation in
figure 39 suggest a strong inverse relationship between approach velocity and t. However, CFD tests indicated that t did not decrease with approach
velocity at the velocities anticipated at full scale. To systematically verify
this, a CFD study was implemented on a full-scale bridge model with a clear
span and with an upstream channel and bridge opening width of 40 ft, hb of 7.87 ft, and a bridge
superstructure height of 3 ft. The superstructure configuration was scaled up
from the six-girder model in figure 32.
A range of simulations were completed by varying both the approach depth and
velocity, as shown in
figure 42.
Except at the lower end of the ht/T ratio, the three velocity relationships are virtually identical. To address
this, a reference velocity representative of the experimental datasets was used
in the first two terms of the equation in
figure 39 and then combined with the constant K,
eliminating the strong velocity dependence. This revised value of K was approximately 0.2. The result of
these two changes is shown in figure 43.

Figure 42. Graph. t in CFD tests.

Figure 43. Equation. Modified best fit equation for t.

Finally, the combined coefficient, K, was adjusted upward to provide a degree of conservatism to the
model for design. The appropriate degree was defined as a reliability index of 2,
where reliability index is defined by the equation in figure 44.

Figure 44. Equation. Reliability index.

Where:

β = Reliability index.

μz = Mean of the
variable Z.

σz = Standard
deviation of the variable Z.

The variable Z is
defined as the ratio of the measured scour depth to the calculated scour depth.
The resulting design equation is provided in
figure 45.

Figure 45. Equation. Proposed design equation for t.

This design equation for t is plotted in
figure 42 for comparison with the CFD results for an unscoured full-scale bridge. The
design follows the same general shape as the CFD observations while providing the
desired conservatism as measured by the reliability index.

CFD modeling was used to evaluate the location of the
stagnation point. Thus far, it has
been assumed that the stagnation point for fully submerged flow is at the top
of the bridge superstructure. CFD simulations of fully inundated decks showed
that the stagnation point was generally near the mid-point of the superstructure.
This is shown in
figure 46 for the case of a rigid unscoured bed and in
figure 47 for a noncohesive bed material after scour occurs.

The bridge superstructures modeled using CFD had large
openings on the parapets simulating open bridge rails. Not all bridges will
have such openings, or, if they do, the openings may be smaller. For bridges
with smaller or no side openings above the pavement level, the stagnation point
may be higher than indicated in the CFD analyses.

The bridge models used by Arneson and Abt as well as Umbrell et
al. do not have parapet openings as the TFHRC models do, which may affect the
stagnation streamline location.(1,2) Structure design variations create
differences in results. With the data available, it is not yet possible to draw
definitive conclusions about the stagnation point location. Therefore, the
assumption that it is at the top of the bridge superstructure is maintained.

Combining the general equation for scour in
figure 14 with the equation for t in
figure 43 yields the best fit scour equation in
figure 48.
Alternatively, using the t equation
in
figure 45 yields the proposed design equation in
figure 49.
Both equations are for fully or partially submerged bridge flow under clear
water scour conditions.

Figure 48. Equation. Best fit equation for submerged bridge flow.

Figure 49. Equation. Design equation for submerged bridge flow.

A performance evaluation of the best fit and proposed design
equations, including comparison with the Arneson and Abt method and with the
method of Umbrell et al., is appropriate.(4,2) However, the Arneson
and Abt and Umbrell et al. methods were based on analyses to provide best fits
to the data being considered at the time. For this comparison, they were adjusted
to consider a conservative envelope for design to directly compare with the
proposed method. An optimization parameter,
β, was added to the Arneson and Abt equation (from
figure
1)
to evaluate and compare its performance as shown in
figure 50.(1)

The best fit equation from
figure 48 was applied to the experimental data and is summarized in
figure 52.
These data result in a root mean square error (RMSE) of 0.070.

Figure 52. Graph. Scour comparison with best fit equation.

A comparison of the best fit model with the Arneson and Abt
as well as Umbrell et al. equations is summarized in
table
1
.(1,2) The table includes the previously defined reliability index. The best fit and
Arneson and Abt models exhibited a low reliability index, which is to be
expected for best fit equations. The Umbrell et al. equation showed a higher
reliability index, suggesting that some degree of conservatism was included in
the model.

The RMSE of the 109 data points is also provided in table 1,
along with the probability that the calculated estimate over predicts the
observed scour. Both of these metrics are based on the dimensionless ratio of ys/hu.

The proposed design model was compared to the Arneson and Abt
and Umbrell et al. models.(1,2) To provide a common basis for
comparison, the parameters β and α
were optimized so
that the reliability index was close to 2, which was also performed when
developing the proposed design model. The comparative metrics are summarized in
table
2.
With a reliability index of 1.9 for all models, the probability of
overprediction ranged from 97 to 99 percent.

Table 2. Comparison of performance for design models.

Comparative Metric

Best
Fit (figure 49)

Arneson
and Abt Model (β =
0.44)

Umbrell
et al. Model (α = 0.06)

Reliability index

1.9

1.9

1.9

RMSE of all observations

0.153

0.457

0.136

RMSE of all over
predictions

0.155

0.451

0.136

Probability of over
prediction (percent)

97

97

99

The RMSE measures complement the reliability index. Given the
equivalent reliability index and probability of over prediction, a lower RMSE
is desirable. Based on RMSE, the Umbrell et al. model performs slightly better
than the proposed model, with
equal to 0.06. The
Arneson and Abt model performs significantly worse than either of the other two
formulations.(1)

Performance is graphically represented in
figure 53 for the proposed model,
figure 54 for the Arneson and Abt model, and
figure 55 for the Umbrell et al. model.(1,2) The graphs for the proposed model
and the Umbrell et al. model are similar, as indicated by the RMSE values.

Figure 53. Graph. Performance of proposed model.

Figure 54. Graph. Performance of Arneson and Abt model (
(β = 0.44).

Figure 55. Graph. Performance of Umbrell et al. model (α = 0.06).

Alternative methods of optimizing each model might be
explored. However, the robustness of a model for application to data not used
for its development depends on sound formulation based on physical principals. The
Arneson and Abt model is primarily regression-based and does not perform as
well as the Umbrell et al. model and proposed model on the combined laboratory
data sets. The Umbrell et al. model is conceptually based on an analog to
horizontal contraction scour. The proposed model is soundly based on physical
principles as previously described. While the Umbrell et al. model performs
slightly better when considering the RMSE for design, the physical basis of the
proposed model should provide for robust application to field applications.

This study was performed to develop a new model for
estimating pressure flow (vertical contraction) scour under partially or fully
submerged bridge superstructures. Based on this model, a proposed design
equation for bridge foundation design for pressure scour is proposed. As part
of this study, a set of laboratory experiments supplemented with PIV tests and
CFD modeling were conducted. These data, combined with those from previous
research efforts, were used to develop and validate the proposed scour model in
clear water conditions with non-cohesive bed materials.

The laboratory experiments showed that under clear water
conditions, the maximum scour occurred at the downstream end of the bridge
deck. The maximum scour depth increased with deck inundation but decreased with
increasing sediment size. It was also observed that the maximum scour depth was
independent of the number of deck girders.

Scaled flume experiments, PIV tests, and CFD modeling confirmed
that the maximum scour depth can be described by the effective depth
(contracted depth) in the bridge opening at equilibrium scour. In order to evaluate
the effective depth, it is necessary to estimate the boundary layer separation
thickness from the leading edge of the bridge deck. When the bridge
superstructure was lowered into flowing water, the leading edge developed a
flow separation area. When the lower cord of the girder approached the bed or when
the deck approached the water surface, the flow separation zone below the
bridge deck was influenced by the bottom boundary and water surface,
respectively.

Aided by a parametric CFD study, a design equation
calculating t was developed and
incorporated into an overall pressure (vertical contraction) scour model for
design. Important features of this model include the following: